![]() |
Mathbox for Alan Sare |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > un0.1 | Structured version Visualization version GIF version |
Description: ⊤ is the constant true, a tautology (see df-tru 1537). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
un0.1.1 | ⊢ ( ⊤ ▶ 𝜑 ) |
un0.1.2 | ⊢ ( 𝜓 ▶ 𝜒 ) |
un0.1.3 | ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) |
Ref | Expression |
---|---|
un0.1 | ⊢ ( 𝜓 ▶ 𝜃 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | un0.1.1 | . . . 4 ⊢ ( ⊤ ▶ 𝜑 ) | |
2 | 1 | in1 44004 | . . 3 ⊢ (⊤ → 𝜑) |
3 | un0.1.2 | . . . 4 ⊢ ( 𝜓 ▶ 𝜒 ) | |
4 | 3 | in1 44004 | . . 3 ⊢ (𝜓 → 𝜒) |
5 | un0.1.3 | . . . 4 ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) | |
6 | 5 | dfvd2ani 44016 | . . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃) |
7 | 2, 4, 6 | uun0.1 44211 | . 2 ⊢ (𝜓 → 𝜃) |
8 | 7 | dfvd1ir 44006 | 1 ⊢ ( 𝜓 ▶ 𝜃 ) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1535 ( wvd1 44002 ( wvhc2 44013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-vd1 44003 df-vhc2 44014 |
This theorem is referenced by: sspwimpVD 44352 |
Copyright terms: Public domain | W3C validator |